### A convenient setting for differential geometry and global analysis II

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Let $\Omega $ be a bounded open subset of ${\mathbb{R}}^{n}$, $n>2$. In $\Omega $ we deduce the global differentiability result $$u\in {H}^{2}(\Omega ,{\mathbb{R}}^{N})$$ for the solutions $u\in {H}^{1}(\Omega ,{\mathbb{R}}^{n})$ of the Dirichlet problem $$u-g\in {H}_{0}^{1}(\Omega ,{\mathbb{R}}^{N}),-\sum _{i}{D}_{i}{a}^{i}(x,u,Du)={B}_{0}(x,u,Du)$$ with controlled growth and nonlinearity $q=2$. The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set ${\mathcal{M}}_{sa}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator ${L}_{\xi}$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp\left(\xi \right)={e}^{i{L}_{\xi}}$, is continuous but not differentiable. The same holds for the Cayley transform $C\left(\xi \right)=({L}_{\xi}-i){({L}_{\xi}+i)}^{-1}$. We also show that the unitary group ${U}_{\mathcal{M}}\subset L\xb2(\mathcal{M},\tau )$ with the strong operator topology is not an embedded submanifold...

We prove that the Quasi Differential of Bayoumi of maps between locally bounded F-spaces may not be Fréchet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F-spaces here are not necessarily locally convex

In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into $H\times [0,+\infty )$, where $H$ is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in $H\times \left\{0\right\}$. Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class $\infty $ with smooth boundary.

Let M be a separable ${C}^{\infty}$ Finsler manifold of infinite dimension. Then it is proved, amongst other results, that under suitable conditions of local extensibility the germ of a ${C}^{\infty}$ function, or of a ${C}^{\infty}$ section of a vector bundle, on the union of a closed submanifold and a closed locally compact set in M, extends to a ${C}^{\infty}$ function on the whole of M.

We first generalize the classical implicit function theorem of Hildebrandt and Graves to the case where we have a Keller ${C}_{\Pi}^{k}$-map f defined on an open subset of E×F and with values in F, for E an arbitrary Hausdorff locally convex space and F a Banach space. As an application, we prove that under a certain transversality condition the preimage of a submanifold is a submanifold for a map from a Fréchet manifold to a Banach manifold.

Differential forms on the Fréchet manifold $\mathcal{F}(S,M)$ of smooth functions on a compact $k$-dimensional manifold $S$ can be obtained in a natural way from pairs of differential forms on $M$ and $S$ by the hat pairing. Special cases are the transgression map ${\Omega}^{p}\left(M\right)\to {\Omega}^{p-k}\left(\mathcal{F}(S,M)\right)$ (hat pairing with a constant function) and the bar map ${\Omega}^{p}\left(M\right)\to {\Omega}^{p}\left(\mathcal{F}(S,M)\right)$ (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians [6].